Volunteers carry an injured person following Syrian government forces airstrikes in Aleppo. (AFP)

Aleppo, Syria: The largest hospital in rebel-held east Aleppo was bombed on Saturday for the second time in days as Syrian government forces pressed a Russian-backed offensive to retake the entire city.

Aleppo, once Syria's vibrant commercial powerhouse, is now at the heart of a major military campaign by President Bashar al-Assad's fighters and his steadfast ally Moscow.

The offensive, announced on September 22, has seen dozens of civilians killed and residential buildings flattened in the east, where an estimated 250,000 people live under government siege.

As the situation for civilians grows increasingly dire, the largest hospital in east Aleppo was hit by two barrel bombs on Saturday, the medical organisation that supports it said.

"Two barrel bombs hit the M10 hospital and there were reports of a cluster bomb as well," said Adham Sahloul of the Syrian American Medical Society (SAMS).

Sahloul said a small group of patients and doctors "were inside the hospital for basic triage, bandaging, and cleaning services for emergency cases" when the bombardment began and remain trapped there.

SAMS radiologist and hospital administrator Mohammad Abu Rajab made an urgent call for help on Saturday morning from inside M10.

"The hospital is being destroyed! SOS, everyone!" he said in an audio message distributed to journalists.

M10 had already been hit on Wednesday along with the second-largest hospital in the area, known as M2, in what UN chief Ban Ki-moon denounced as "war crimes."

That bombardment heavily damaged the two facilities and left only six fully-functional hospitals in the city's east, according to SAMS.

'Flood of violence'

At the bombed hospital, an AFP journalist saw blood-stained hospital beds and dented equipment lying in disarray beneath blown-out windows, while medical staff in green scrubs picked through the rubble outside to assess the damage.

"A new barrel bomb fell this afternoon in front of the hospital, forcing medical staff... to evacuate all patients to another one and leave the hospital," a doctor at M10 told AFP.

French Foreign Minister Jean-Marc Ayrault denounced the strikes, saying that "the systematic targeting of structures and health workers is particularly unjustifiable".

"This new attack only confirms the absolute urgency of a cessation of hostilities in Aleppo and access for civilian populations to the humanitarian assistance they desperately need."

The World Health Organization has called Syria the most dangerous place in the world for health workers, and Aleppo in particular has seen much of its medical infrastructure destroyed or heavily damaged.

Since fighting first broke out there in 2012, Aleppo has been divided by a front line between rebel forces in the east and government troops in the west.

After the government launched its offensive last month, more than 220 people have been killed by bombardment on Aleppo's east, including six children and 12 other civilians on Friday, according to the Syrian Observatory for Human Rights.

In west Aleppo, rebel rocket fire killed 15 civilians and wounded 40 on Friday, state television reported. The official news agency SANA said 13 people were wounded Saturday in the western Al-Midan neighbourhood, also by rebel shellfire.

The assault has seen government forces seize territory in both the Suleiman al-Halabi neighbourhood in the city centre and on the northern edges of Aleppo.

'Bombs are raining'

On Saturday, regime loyalists backed by heavy raids advanced on the edges of the Bustan al-Basha neighbourhood in Aleppo's north, the Observatory said.

An AFP correspondent said clashes and the loud booms of shelling were heard around the Suleiman al-Halabi and Bustan al-Basha fronts throughout the night.

Residents of regime-held areas expressed relief that the rebels were being pushed back, but said they feared retaliation.

"We were happy when we heard about the army's advance," said Majed Abboud, a 32-year-old car dealer.

"But I'm afraid that with these ferocious clashes, there will be some kind of reaction from the armed groups."

The battle for Aleppo has sparked some of the most brutal violence since the March 2011 beginning of Syria's conflict, which has killed more than 300,000 people and displaced over half the population.

The Doctors Without Borders (MSF) medical charity warned Friday that "bombs are raining" over the city, turning east Aleppo into "a giant kill box".

Diplomatic efforts to end the fighting across the country have all but collapsed.